WAVE FUNCTIONS OF A FREE ELECTRON IN AN EXTERNAL FIELD AND

J , BERGOU S , VARRÓ

K FKI-1979-70

WAVE FUNCTIONS OF A FREE ELECTRON IN AN EXTERNAL FIELD AND

THEIR APPLICATION

IN INTENSE FIELD INTERACTIONS. II.

RELATIVISTIC TREATMENT

’H u n g a r i a n ‘A c a d e m y o f S c i e n c e s

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

2017

WAVE FUNCTIONS OF A FREE ELECTRON IN AN EXTERNAL FIELD AND THEIR APPLICATION IN INTENSE FIELD INTERACTIONS, II,

RELATIVISTIC TREATMENT

J. Bergou and S. Varró

Central Research Institute for Physics H-1525 Budapest, P.O.B. 49, Hungary

Submitted to J. Phye* A.

HU ISSN 0368 5330 ISBN 963 371 593 6

ABSTRACT

The behaviour of a relativistic free electron in an external plane wave field is analysed and a review of the existing solutions of the correspond­

ing Dirac equation is presented. Completeness and orthogonality of the Volkov states are also proved. Based on the exact wave function obtained a relati­

vistic generalization of the perturbation method proposed in a previous paper is elaborated as a means of treating intense field problems in a covariant manner.

АННОТАЦИЯ

Анализировано поведение релятивистского свободного электрона во внешнем плоском волновом поле, и представлен обзор различных точных решений соответ­

ствующего уравнения Дирака. Обсуждены комплектность и ортогональность состо­

яний Волкова. На основе точной волновой функции разработано обобщение метода возмущения, предложенного в изданной ранее работе для обсуждения проблем ин­

тенсивных полей при помощи ковариантного метода.

KIVONAT

Relativisztikus szabad elektron külső sikhullámtérbeli viselkedését ana­

lizáltuk a megfelelő Dirac egyenlet különböző egzakt megoldásainak áttekinté­

sével. Diszkutáljuk a Volkov állapotok teljességét és ortogonalitását. Az eg­

zakt hullámfüggvényre alapozva egy előző cikkben javasolt perturbációs mód­

szer általánositását dolgoztuk ki intenziv térbeli problémák kovariáns tár­

gyalására .

In a previous paper we have presented a review of the solu­

tions of the nonrelativistic free electron external field inter­

action problem /Bergou, 1979/. We have shown the equivalence of some at least superficially different solutions and proposed a perturbation method to treat scattering problems in the presence of an intense external field. In this method we used the complete set of the exact wave functions of the free electron in the field as a basis and treated the scattering potential as a p e r t u r b a ­ tion. In the present paper we give a similar account of some ex­

isting solutions of the corresponding Dirac equation, prove their equivalence, orthogonality and completeness and, using this com­

plete set of relativistic w a v e functions, we give a simple gener­

alization of the above mentioned perturbation method and de t e r ­ mine the validity of the dipole approximation as well as the

validity of the nonrelativistic Born-approximation in the present problem.

The exact solution of the Dirac equation of a relativistic free electron in an electromagnetic plane wave field has long been known /Volkov, 1935/. This famous result has, since that

time, been reproduced by several authors using different methods.

It was shown, for example, that this problem can also be solved by purely algebraic methods /Beers and Nickle, 1972/. In another paper the so-called projection technique led to the same result

/Becker and Mitter, 1974/ . The Dirac equation, however, can also be solved without the direct use of the special assumptions and

specific methods applied in these papers. By choosing an appro­

priate coordinate system the system of the coupled differential

2

equations for the spinor components can be reduced to an o r d i ­ nary first order differential equation for each component separ­

ately if one uses Majorana representation instead of the stan­

dard representation for the Dirac matrices. In this context it is interesting to mention another method /Alperin, 1 9 4 4 / . It is of course well known that in the "derivation" of the Dirac e q u a ­ tion the originally irrational Hamiltonian /given by a square root expression/ is rationalized by the usual Dirac m a t r i c e s . The basic idea of Alperin's paper was to exploit the symmetry of the problem by a suitable choice of coordinates, ensuring that both the rational and irrational parts show the required symmet­

ry. Based on the wave function such obtained he determined the scattering cross section of an arbitrarily intense classical e,m.

field by an electron using the method of the transition currents.

The paper did not, however, draw much attention at that time, nor s i n c e .

The next section is devoted to the orthogonality and com­

pleteness problem of the Volkov states - this being the central problem in perturbation theoretical applications. In Section 3, the solution in Majorana representation and rederivation of the Alperin solution are given and their unitary equivalence with the Volkov solution is proved. In Section 4, it is shown how the m u l ­

tiphoton radiative corrections to the scattering of a free e l e c ­ tron on a background potential due to the interaction with an in­

tense mode of the e.m. field /laser/ can be obtained by using the Volkov states. In the last section we deal with the connection of the present approach with the method introduced by one of us in a previous paper. The limits of validity of the nonrelativistic dipole approximation as well as other consequences of the relati­

vistic generalization, are also discussed.

2 . THE VOLKOV STATES

In an external electromagnetic field characterized by the A /х/ four-vector potential the relativistic wave equation of a

spinor electron has the form /cf.Bjorken and Drell, 1964; we shall use the metric and notation as well as representation of

the Dirac matrices of this r e f erence/:

(ifl-eA-u)'i' = О /2.1/ where

e

e - ЬБ ^/2.1а/

/here c is the velocity of light, ft is Planck's constant divided by 2п / . We choose A /х/ as representing a transverse plane wave, i .e.:

The well-known positive and negative frequency Volkov type s o l u ­ tions of the above Dirac equation represent modulated plane waves, where the modulation depends only on £. The plane wave itself can be parametrized by the 4-momentum lying on the free mass shell

/initial conditions are not taken into a c c ount/:

A(x) = A(E) , £ = k* x к •A = к 2 = О /2.2/

In the case of a general elliptically polarized wave

Ais) — e-^A^(E)+e 2 ^ 2 ( E)

/2 .2a/

k*e^ = 0, e^*ej &± . / i = 1, 2/

/2.3/

where

0 , p 2 = и 2 , py = (|p0 l, £) /2.3a/

and

Jp ±)(S) = ^ p [ ± 2ep*A(i)-e2A 2 (E)] /2.3b/

One can follow the method of obtaining the solution in this

covariant form in a paper by Brown and Kibble /Brown and Kibble, 1964/ .

4

These Volkov states were applied by several authors to treat the interaction of a free electron with an intense optical mode /ac­

counted for by the external field approxim a t i o n / . Interaction with another weak mode or some other weak potential can be taken into account by the usual perturbation theory. By this method ab­

sorption and emission from the intense mode can be directly com­

puted up to any order contrary to the usual Feynman-Dyson ap­

proach .

For the E ^ ^ C x ) matrices introduced in /2.3/ one can easily P

verify the following relationships to hold /Ritus, 1972/:

d 1*x E ^ ± ^(x)E^± )(x) = (p-q) »

J Er

E ^ W E ^ C y ) = 6 (')(x -y) where

E y°e+y°

Orthogonality and completeness given in this form are not satis­

factory for our purposes since the 4-momentum components are not on the free mass shell. Therefore in the following treatment we give different orthogonality and completeness r e l a t i o n s . For further investigation of the Volkov states it is convenient to use the light-like components originally introduced by Neville and Rohrlich /Neville and Rohrlich, 1971a and b; see also Becker and Mitter, 19 74/ . This formalism is based on the fact that the vectors

nu = k u = M l , n), co\/2 s/2

nu = ±-(l, -n), e. = (0, e.) i=l, 2 . Ал"

72.4/

form a complete orthonormal set in Minkowski space, therefore any "a" four-vector can be given by its light-like components in the following way

a = au n + av n + + a2e2 /2.4а/

where

a.. = n*a , a„ = n*a , a^ = -a'e^ i=l, 2

u v / 2 .4b/

Taking into accoung /2.4/ - / 2 . 4 b / the solution /2.3/ can be brought to the form

t(±)(uvxí ) = YvYia1 (u)]u^±)-

• e"i[upu+vpv"xip i+jfp± ^ u ^dul

/2.5/

where

u » x f , v = xu , A^(£)= a., (u) 1=1, 2 /2.5а/

and

fp+ ^(u) = 2| - [ + 2 p 1ai (u)+eai (u)a1 (u)]

r ' T '

The states /2.5/ /with normalization factor --- Ц т т(^~) form (2n)3'2 P ”

an orthogonal set in the sense x 7

12.5b/

1/2

JVpr ^ ( u v x i ) YvVp+f I f (uvx± ) d v d 2 x± =

= 6(pv - p ; ) 6 (2)(P i - p p 6 r r /

T ^ + ^ (u v x ± )Y £,(uvx±)d v d 2 x± = О ,

pr p

W 2 W+Y°

/2^.6^/

(2)

Here r = 1, 2 are the spin indexes and 6 4 7 denotes the two-di­

mensional Dirac-delta function. The normalization of the u^~^ bi spinors is as usual

? ^ u (±> - ±1

p p /2.6а/

б

То obtain the appropriate definition of completeness we deal in the first step with the completeness of free plane waves. The solutions of the Dirac equation of a free particle are:

(±) , * 1

Ф ' (x) pr

/И (+) +i p.x

V е

The definition and the normalization condition of the u (±) bi- pr

spinors are again given by Eqs./2.3a/ and /2.6а/. The complete­

ness relation of the set of positive and negative frequency solu­

tions is r=l,2

l

d 3p cp( + ) ( х ) ф ^ (х')+ф ^ ^ (х)? ^ (x')

pr pr pr pr X =x

о о Here we made use of the fact that

v (±) f±] jó±h К г -pr pr = К

=x'V=0(3)

(2.8)

(2.8a)

Relation /2.8/ can be generalized in a covariant manner such that instead of the x Q =const 3-space we define completeness on a space like hyperplane determined by an arbitrary timelike normal vector For the symmetry of the external plane wave field the best choice is the u=const null-plane. Therefore in full analogy with /2.8/

to establish completeness of the Volk o v states on the null plane we investigate the expression

V(x,x') = £ r

, 3 -

d p WÍ+* ( x ) w £ } (x,)+vj^” ) ( x ) ^ " 5 (x')

pr pr pr pr (2.9)

or in particular V(x,x')| ,

'u=u' ^{d p.}

d 2p .(1+|SÍ.)Ét>*(1 = M ) Y (2.9a) 2k*p'2pv 4 2kpy,v

Here and below

00 oo

0 “ ° °

= vpv - x ip i (2.9b) In obtaining /2.9а/ from /2.9/ we used the relation /2.8а/ and changed pv to -pv in the negative frequency term. Before giving the completeness relationship of the Volkov states, we investi­

gate the meaning of the operators defined by /2.9/ and /2.9а/.

Using the /2.6/ orthogonality relations it is easy to prove the validity of the following projection properties

| а 3й ' у ( х , х ' ) ^ ^ ( х ' ) = w(*)(x)

►

| d 3x ' ^ ^ ( x ,)Y0V + (x//x ) Y 0 = W ^ ^ ( x )

/2 .10/

Introducing the bra and ket vector notation, the algebraic mean­

ing of the relations /2.10/ becomes even more apparent in the abstract state vector space:

V ( u , u ' ) | w ^ ) (u')> = | W ^ )(u)>

/2.11/

where

v(u',u) = y°V+ (u/,u)y° /2.11а/

From /2.11/ it is clear that v(u,u') and v(u',u) represent p r o ­ pagators of the Volkov states and the Dirac-adjoint Volkov states, respectively. It is also clear that v(u,u) and V(u,u) are the

projectors of the corresponding states. Other abstract algebraic properties of the Volkov states w i l l be discussed in more detail in a subsequent paper.

Completeness of the Volkov states on a light-like hyperplane taking into account E q s . /2.10/ and /2.11/ can now be expressed by the formula

[V(^x ,^x' ) + Y ° V + (^x',^x )^y° ] u = u , = 6 ( 3 ) ( x - x ' ) -

- ih| e(v-v') 6^2^(x±—x')y v

/2.12/

where the definition of e(v) is given by the integral

.

- 8 -

e(v)

сю 2 [

Ц .

sin VPv p.

dp

V V

1 v>0 0 v=0 -1 v<0

/see also Neville and Rohrlich, 1971a, App II./.

On the basis of /2.12/ we accept the assumption that any bispinor function can be represented as a /generalized/ linear combination in terms of Volkov states .

3 . CONNECTION WITH OTHER SOLUTIONS

It is obvious from the preceding section that the problem of a free spinor electron interacting with an external plane wave field can be solved exactly in a covariant manner by using the light-like formalism. In this section we show two important e x ­ amples where the Hamiltonian form of the corresponding Dirac equation, with appropriate coordinate systems, can also be s o l v ­ ed exactly. Let us chose the у -axis of our coordinate system as

coinciding with the direction of the wave vector of the light field given by the A /х/ vector potential, and polarization p a r a l ­ lel to the x - a x i s . The Dirac equation of the problem in this coordinate system is

- E A x ) +

V 1!?) + +

+ &X]T = 2.— - Ф Эхо

/3.1/

where

A x = A(|E) , E - X0 -y /3.1а/

A is an otherwise arbitrary function of £.

We look for the solution of /3.1/ again in the form of a plane wave modulated by the external field:

W = exp{-i(x0p 0-xpx -ypy -zpz)}®(£) /3.2/

f

I

1

Here ф(£) is a bispinor function for which, after substituting /3.2/ into /3.1/, we get the following ordinary differential equation

{a x [px-eA x (£)] + a yPy +. a zP z + Эк-ро )Ф =

= ( 1 - a

/3.3/

In the representation of the Dirac matrices used throughout the present paper this is a system of coupled equations since ay on the r.h.s. couples the derivatives of the different bispinor com ponents of Ф. It is easy, however, to get rid of this difficulty in Majorana representation /Majorana, 1937/, where the /3.3/ sys tern of equations is decoupled:

[ax (*Px+eA x )+ßPy-azp z+ay >i-p o]®' = (l-ß)ig|- /3.4/

If we introduce ф' and x' the upper and lower components of Ф' resp., /3.4/ gives a simple algebraic equation between ф' and x'

Ф' = p- ^p-[(-P x +eAx^ax -pzaz+MOy / 3 *4a/

о * y

where о -s are 2x2 Pauli matrices.

If /3.4а/ is substituted into the lower component equation of /3.4/ we get

2i3 E “ = < i ^ K P x -eAx )2+p^-H2 ]-pg >x' /3.4b/

where

Pp “ Po'Py ' Pe = P o +Py /3 .4c/

Without loss of generality we can assume that the parameter p satisfies the usual free mass-shell relationship

2 2 2 2 2 p* = p^+p^+p^+x

у z

2 2 2

P Pr - P +P +и

£ * X

/ / 3.4 d /

10

Then from /3.4Ь/

x' = exp[i

V E)

A 2 (E)]

/3.4e/

J p ' (E)

Here Xq is a constant spinor. The final form of the solution of /3.4/ using /3.4а/ then becomes

T ' =

— (-p a -p о +ИО ) x ,+— eA a x'

T~* 4 X X v ~ r r CZ Z \r7 Л*Г* r* у o p X X^O j—i [p*x+|j^+ ^ (E)d£]

/3.5/

The exponent of /3.5/ agrees with that of the Volkov states, while the proof of the equivalence of the bispinor amplitudes will be given in what f o l l o w s . In the special coordinate system

introduced in the above calculation the Volkov-bispinor reads u (v) = ( l + # £ ) u =

p v 2k«py p

eA

l - d + a y K j ^ :

In Majorana representation this becomes eA.

u<v >' =

p l+(l+&)a. x

X 2 p ^ ^u: /3.7/

where now u^ satisfies the transformed free energy eigenvalue- -equation

P 0 u ; " la'E + P ' * ] < - t-axP x + e P y - a zP z +ay H ] u ^ /3.7а/

ion /3.7/ written out

( \ ^- f \ f

0 °x eA

1+ =

0 0 x f

Ko

l J ^J V У \

now eA

cpfH-- x*

v o p^ x ^ o X'ло

/3.8/

On the other hand from /3.7а/

cp' = — (-p a -p a +40 )xA p 4 *x x z у' о

Л

/3.8а/

Substituting /3.8а/ into /3.8/ and comparing the result with /3.5/

one can see that the solution given here is equivalent with the Volkov solution.

Another interesting solution of the present problem was given by Alperin /Alperin, 1944/ . In the following we shall re­

peat with some modification the original derivation of Alperin's solution. We start from the relativistic energy-momentum formula

h - M - e£)2 + к 2 /3.9/

or in operator form

In the special coordinate system used throughout in the preceding calculation it is more convenient to take the square root in a different way, namely

£> -eA

x i y - p ^ ^ l + u 2 /3.10/

where

^ - !y> - k ' \

^

+ У -

/3.10а/

£ = xo-y , Л = x0+y

The matrices = 7(ay +iax ) and = 2-(a.y~iax ) satisfy the com­

mutation relation a ca_+ a a c = -1

5 Л Л 5 /3.11/

By using these matrices the Dirac equation corresponding to /3.10/ has the following rational form:

tfx -eAx )W = i ( a ^ g-KxTlß n+azß 2+ßH)’P /3.12/

The solution can be looked for again with the usual ansatz

e l(xpx+2p z4 E P E4 n p n )e(E) f

/3.13/

P£ = P o +Py' Рл = P o-Py

After substituting /3.13/ into /3.12/ we obtain a coupled system of equations for the components of Ф :

-i [px-eAx (£) ]®1 «* -pT1®4+pz®3+K® 1 /3.14а/

d<Eu -

-i[px -eAx (E)]Ф2 = P £®3+2i-g|~p2®4+H®2 /3.14b/

-i [px -eAx (E) ]Ф3 = -pn®2+Pz® i “H®3 /3.14с/

d®.

-i [px -eAx (E) ]®4 = P g ®1+2i-g|~pz®2-H®4 /3.14d/

From /3.14а/ and /3.14c/ ®2 and Ф4 can be expressed by Ф^ and Ф3

® 2 = {i [ (Px-eA x ) +iw 1ф з+р2ф 1 > /3.15а/

Р Л

Ф4 = ^-{i t (px -eAx )-í h]®1+P2®3 } / 3.15b/

Р Л

and substituting these expressions into /3.14b/ and /3.14d/ we

I

obtain two similar uncoupled equations for Ф^ and Ф3:

21- a H = (5-t(px -eA x )2+p2+H 2 ]-pE )®1>3 / з а « / The solution of /3.16/ taking into account /3.4 d / will be

Ф -L 3 = Фд^ 3 (0)e_ijJp 3 (0) = const /3.16а/

Through /3.15а/ - /3.15Ь/, all four components of ф are known, thus another solution of the Dirac equation is found. The func­

tion in the exponent of this wave function coincides with e x ­ ponents of the Volkov states and the state found in Majorana re presentation. All we have to show is the equivalence of the b i ­

spinor part with the previous solutions. From /3.4а/

-1*1 = i-{[i(p -6A J-KlXjj-tPjiX!) r n

ф2 = i-{[i(px-eAx )+K]iXl+ P zX 2 }

/3.17а/

/ 3 .17b/

/3.18/

As the T matrix defined here is unitary the Alperin type solu­

tions are equivalent with the solutions obtained in the Majorana representation and due to the transitivity property of the uni­

tary transformations the three formally different wave functions considered so far are interrelated by unitary transformations and they are therefore equivalent from the physical point of v i e w .

4 . AN APPLICATI ON OF THE VOLKOV STATES

In a previous paper of one of the present authors /J.B./, the wave functions of a nonrelativistic free electron moving in a homogeneous external field /dipole approximation/ were used as the basis set of a perturbation method to calculate the cross section of the inverse as well as induced multiphoton bremsstrah- lung process /Bergou, 1979/. In this section we work out an ob­

vious generalization of the method for the relativistic case and /primes are omitted for the sake of simpler n o t a t i o n / . Comparing the above relations with /3.15а/ and /3.15b/ we can immediately see that the same relation holds between J 1ср1^ and as bet-

К ) К ) l l * 2J

ween Ф . and Ф0 , therefore if we make the identifications anc^ X2 "* ф 3 corresP onding -itpjL - Ф2 and ф 2 - Ф4 id identification must also hold. From this consideration the con­

nection between Ф = (^) and (ф ) can be written in the following compact form

14

beyond dipole approximation by using the Volkov states as the basis set. Similar problems were touched on earlier /Denisov and Fedorov, 1967; and Brehme, 1971/ where the analytical and the numerical behaviour of the relativistic cross section formulae of the scattering by a Coulomb background were investigated by different methods and in a recent paper /Ehlotzky, 1978/ results beyond dipole approximation but using nonrelativistic descrip­

tion were published.

Consider the problem of the scattering of a relativistic free electron by a V /г/ scalar background potential in the p r e ­ sence of an intense electromagnetic mode /laser light / . The i n ­ tense mode can be accounted for by the external field approxima­

tion and the corresponding Dirac equation reads /using light->

-like formalism/

(Yv í3v+y u í9u-y í [ia-^-Ea-j.(u) ]-е^ (v-u, x í)-k }W = 0 /4.1/

я - 9 a - 9 a - 9 9u~ 3u ' 9v “ 3V ' l

We look for the solution in the form

W = + ) + ш /4.2/

qr к

where q,r are determined by the parameters of the initial state and the correction term is a superposition of the /2.5/ Volkov states taking into account the /2.12/ completeness relation:

00 00

Wk (uvx± ) = I dp

v (uvXl) +

+ c ^ ( u ) W ^ (uvx. ) ] PvPir P r 1

/4.3/

Here c /+/ and c^ ^ are scalar amplitudes to be determined. For the sake of simplicity we choose the initial conditions

' b r 1 <u i V = 0 f o r a l l p a n d r /4.3а/

Upon substitution of /4.3/ into /4.1/ we obtain the equation /in more compact notation/

V

“du +^ u * = e ^ + )+elirwk /4.4/

In first approximation we neglect the second term on the r.h.s.

which contains the product of the correction term and the p e r ­ turbing potential thus giving higher order corrections only.

Then we take the scalar product of the remaining terms with from the left and obtain the following ordinary differen-

Si ■*- I /

tial equation for с^ /и,и^/

d c (+) ŰCq'r'

^ d Z T - d v d 2x i^ y ^ 7 ( u v x 1 ) e y W g ^ (uvx^ /4.5/

Here we directly made use of the /2.6/ orthogonality relations

— P v —

and that for arbitrary spin orientation u v .u_ = — u_u and p v p к P P

UpYvVp = 0. Equation /4.5/ can be integrated in a simple way leading to

u

qt^f(u,ui ) = -ie d u f |dvd2x1W^f^™, qr⁽⁺⁾ /4.5a/

Uj

The transition matrix element of the q r + q ' r ' process is con­

nected to the (u,u') amplitude in the following way 4

T fi = cq^r' (u * Ui ^■) /4.6/

or

fi = -ie

a 4x ^ T | , ?

We perform the calculation for a circularly polarized wave A^ (E) = a cosE , A 2 (£) = a sin£ /4.7/

16

Then from /4.6/

fi -i e d 4x U q ,r ,(l-ím r)Y°(l+5 Ö lr)u„r . 2q • k.' qr

• V(x)ei[(g,"^),X+2 Sln(kX“X)]

/4.8/

^x^+a0 ,~T

l2

q ' • e a l,2 e a ^ ~2q'-k

1,2 _ f l , 2 , 2q*k ' ' a.

sin x = — ,

2 2 ,e а , q = q+^n--- к

^ 2k*q

/4.8а/

To evaluate the integration in /4.8/ we use the well-known Fourier expansion

iz sincp г J (z)e

i = l n

n = - o o

incp

Jn denotes a Bessel function of integer order. Thus the transi­

tion matrix element represents an infinite sum of photon absorb­

ed and photon emitted terms

oo

Tf i “ I ТЙ > - Tf i ) =

~2ni t f i > 6 ( 3 i- 5 o+nk) /4-9/

(П) _

fi s V V (V r Mn uq r ^ Q = q'-q+nk ,

“n v(Q n } = d 3x V(x)e i^n -

/4 .9a/

Relation /4.9/ expresses, in an explicit way, energy conservation, the wavy line stands for the fact that electron energies in the presence of the external field are different from those of a bare electron /see e.g. the last of the /4.8а/ relations/. In /4.9а/

M n = (Yo+'4q‘ f ^ v K + (||-YoYv +

n V * V ^ v

/4.10/

ea \

+ 2 ^ r-Yv Yo )

^ v

(+)c + é (" }c n-1 * cn+l

where

n

, , -inx (+) Jn (z)e , e

2 (el"ie2)

(-) / 4 . l o a /

After averaging over initial and summing up for final spin v a r i ­ ables one obtains finally for the scattering cross section

da (n) dfl

, I 0 da^n ^ .. q q' + q q.' q' q f " J n (z)— der 7<1+--- Г2---- > Ч Г > +

и (n)

^ ~ | Q - ( i r ) ( a n V + 0 n v 2 + Y n v 3 + 6 n v 4 )

/4.11/

where

da.(n) В

dfi 2 H E V <5„> V = ^ea

и v 2 = 8»10-11X 2I /4.11а/

The parameters a n , 8n , Yn and depend upon the four-momentum of the electron as well as on the frequency and polarization of the light field but they are almost independent of light intensity.

Their analytical expression is rather complicated and ddes not give a better insight into the physical process involved there­

fore we omit them here. In /4.11/ the first term is just the g e n ­ eralization of the result obtained by the nonrelativistic dipole approximation, while the second term comes from the interaction of the spin momentum with the e.m. field and is exact in the

sense that in all orders it is given by a fourth order polynomial of the intensity parameter v, only the coefficients being slowly dependent on the order of the process and intensity.

5 . DISCUSSION AND SUMMARY

As is well known, an intense mode of the electromagnetic field can be represented by a c-number plane wave field. The central problem of the semiclassical theory is, therefore, the solution of the wave equations of charged particles in such a surrounding. As an extension of previous work /Bergou, 1979/ on

18

exact wave functions, in the present paper we have given a de­

tailed study of the Volkov states from some special aspects.

In Section 2 using the light-like formalism we have shown that the Volkov states paremetrized by the four-momentum on the free mass-shell form a complete orthonormal set on the k-x = const n u l l - p l a n e . The orthogonality and the completeness of this kind are consequences of the special symmetry of the external plane wave field, i , e . of the dependence of the vector potential on the quantity k-x only. As several authors have made direct use of these solutions in perturbation theoretical calculations of d i f ­ ferent kinds, it seemed to us to be important to prove the com­

pleteness of this system and to examine in what sense they can be applied as a basis set.

In the next section we gave two simple methods for the solu tion of the Dirac equation under consideration, each of the

methods was based on the fact that with appropriate choice of the coordinate system the coupled system of equations for the bispinor components can be decoupled into ordinary differential equations for each component separately in a suitable representa tion for the Dirac m a t r i c e s .• This was first performed in the Majorana representation and another solution was found by a suit

able rationalization of the relativistic energy-momentum formula.

We note here that neither of these two methods of solution r e ­ quired the solution of a second order equation as was done in the original derivation by Volkov. We.have shown that the b i ­ spinor amplitudes of the solutions found in this way are related to the Volkov amplitudes through unitary transformations /the agreement of phases is obvious/ and consequently they are equiv alent with each other from the physical point of view.

In the last section the use of the Volkov states was d e ­ monstrated in the derivation of the nonlinear inverse and i n ­ duced bremsstrahlung scattering cross section. The expression ob tained can be considered as a relativistic generalization of the results obtained in the n ^ r e l a t i v i s t i c dipole approximation.

Scattering is elastic with respect to the background potential and inelastic with respect to the external field. This last p r o ­ perty is expressed by the Bessel functions, while corrections to this result were found from two different origins. The first is

what one would expect when nonrelativistic dipole approximation is dropped /relativistic non-dipole part/ and the second comes

from the relativistic interaction of a spin momentum with an e x ­ ternal field. It is interesting to note at this point that this second correction is given by similar finite /fourth order/

polynominal of the intensity parameter in all orders, the coeffi­

cients of the polynomial being only slowly dependent on the order and intensity. From here we may conclude that in a sufficiently intense external field, relativistic effects may become important.

REFERENCE

Alperin M 1944 ZhETF L 4 , 3

Becker W and Mitter H 1974 J. P h y s . A: Math., Nucl. Gen., Т_,12Ъ6 Beers В and Nickle H H 19 72 J. Math. Phys. 1_3, 1592

Bergou J 1979 J. Phys. A: Math., Nucl. Gen., submitted for p u b l . Bjorken J D and Drell S D 1964 Relativistic Quantum Mechanics

/McGraw-Hill Book Co., New Y o r k / Brehme H 1971 Phys. Rev. C 2* 831

Brown L S and Kibble T W В 1964 Phys. Rev. 133A, 705 Denisov M M and Fedorov M V 1967 ZhETF 52/ T34o

Ehlotzky F 19 78 Opt. Comm. 21_, 65 Majorana E 1937 Nuovo Cimento 14, 171

Neville R A and Rohrlich F 19 7ИГ Nuovo Cim. 1A, 625 Neville R A and Rohrlich F 1971b Phys. Rev. D 3, 1692 Ritus V I 1972 Annals of Physics 69_, 555

Volkov D M 1935 Z. f. Phys. 94, 250

С?'l- ^4yfV 1

Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Krén Emil

Szakmai lektor: Zawadowski Alfred Nyelvi lektor: Harvey Shenker Példányszám: 60 Törzsszám: 79-832 Készült a KFKI sokszorosító üzemeben Budapest, 1979 . október hó